User:Roeles

Something for the speed-to-fly page later on.

Known variables: $a,b,c,V_{thermal},S_{g}lide$ Unknown variable: $V_{glide}$

$P(v)=av^{2}+bv+c$

$t_{total}=t_{glide}+t_{thermal}$

$t_{glide}={\frac {S_{glide}}{v}}$

$t_{thermal}={\frac {S_{thermal}}{V_{thermal}}}$

$S_{thermal}=P(v)*t_{glide}=P(v)*{\frac {S_{glide}}{v}}$

Failed to parse (Conversion error. Server ("/media/api/rest_") reported: "Cannot get mml. upstream connect error or disconnect/reset before headers. reset reason: connection termination"): {\displaystyle t_{thermal}={\frac {P(v)*{\frac {S_{glide}}{v}}}{V_{thermal}}}}

$t_{total}={\frac {S_{glide}}{v}}+{\frac {P(v)*{\frac {S_{glide}}{v}}}{V_{thermal}}}$

$t_{total}=S_{glide}*({\frac {1}{v}}+{\frac {P(v)*{\frac {1}{v}}}{V_{thermal}}})$

We want to find the minimum of this formula. Since $S_{glide}$ only acts as a scaling factor, we can eliminate it:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle t_{total} = \frac{1}{v} + \frac{P(v) * \frac{1}{v}}{V_{thermal}}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle t_{total} = \frac{1}{v} + \frac{P(v)}{v*V_{thermal}}}

$t_{total}={\frac {1}{v}}+{\frac {av^{2}+bv+c}{v*V_{thermal}}}$

$t_{total}={\frac {V_{thermal}}{v*V_{thermal}}}+{\frac {av^{2}+bv+c}{v*V_{thermal}}}$

$t_{total}={\frac {av^{2}+bv+c+V_{thermal}}{v*V_{thermal}}}$

$t_{total}={\frac {a}{V_{thermal}}}*v+{\frac {b}{V_{thermal}}}+{\frac {c+V_{thermal}}{v*V_{thermal}}}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle t_{total} = \frac{a}{V_{thermal}}*v + \frac{b}{V_{thermal}} + \frac{c+V_{thermal}}{V_{thermal}} * \frac{1}{v}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle t_{total}' = \frac{a}{V_{thermal}} + \frac{c+V_{thermal}}{V_{thermal}} * \frac{1}{v^2}}

Failed to parse (Conversion error. Server ("/media/api/rest_") reported: "Cannot get mml. upstream connect error or disconnect/reset before headers. reset reason: connection termination"): {\displaystyle {\frac {a}{V_{thermal}}}+{\frac {c+V_{thermal}}{V_{thermal}}}*{\frac {1}{v^{2}}}=0}

${\frac {c+V_{thermal}}{V_{thermal}}}*{\frac {1}{v^{2}}}={\frac {-a}{V_{thermal}}}$

${\frac {1}{v^{2}}}={\frac {-a}{V_{thermal}}}*{\frac {V_{thermal}}{c+V_{thermal}}}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \frac{1}{v^2} = \frac{-a}{c+V_{thermal}}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle v^2 = \frac{c+V_{thermal}}{-a}}

$v={\sqrt {\frac {c+V_{thermal}}{-a}}}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle v = \sqrt{\frac{-c-V_{thermal}}{a}}}